The $\boldsymbol{\beta}$-model for random graphs is commonly used for representing pairwise interactions in a network with degree heterogeneity. Going beyond pairwise interactions, Stasi et al. (2014) introduced the hypergraph $\boldsymbol{\beta}$-model for capturing degree heterogeneity in networks with higher-order (multi-way) interactions. In this paper we initiate the rigorous study of the hypergraph $\boldsymbol{\beta}$-model with multiple layers, which allows for hyperedges of different sizes across the layers. To begin with, we derive the rates of convergence of the maximum likelihood (ML) estimate and establish their minimax rate optimality. We also derive the limiting distribution of the ML estimate and construct asymptotically valid confidence intervals for the model parameters. Next, we consider the goodness-of-fit problem in the hypergraph $\boldsymbol{\beta}$-model. Specifically, we establish the asymptotic normality of the likelihood ratio (LR) test under the null hypothesis, derive its detection threshold, and also its limiting power at the threshold. Interestingly, the detection threshold of the LR test turns out to be minimax optimal, that is, all tests are asymptotically powerless below this threshold. The theoretical results are further validated in numerical experiments. In addition to developing the theoretical framework for estimation and inference for hypergraph $\boldsymbol{\beta}$-models, the above results fill a number of gaps in the graph $\boldsymbol{\beta}$-model literature, such as the minimax optimality of the ML estimates and the non-null properties of the LR test, which, to the best of our knowledge, have not been studied before.

Causal inference of exact individual treatment outcomes in the presence of hidden confounders is rarely possible. Recent work has extended prediction intervals with finite-sample guarantees to partially identifiable causal outcomes, by means of a sensitivity model for hidden confounding. In deep learning, predictors can exploit their inductive biases for better generalization out of sample. We argue that the structure inherent to a deep ensemble should inform a tighter partial identification of the causal outcomes that they predict. We therefore introduce an approach termed Caus-Modens, for characterizing causal outcome intervals by modulated ensembles. We present a simple approach to partial identification using existing causal sensitivity models and show empirically that Caus-Modens gives tighter outcome intervals, as measured by the necessary interval size to achieve sufficient coverage. The last of our three diverse benchmarks is a novel usage of GPT-4 for observational experiments with unknown but probeable ground truth.

The causal revolution has stimulated interest in understanding complex relationships in various fields. Most of the existing methods aim to discover causal relationships among all variables within a complex large-scale graph. However, in practice, only a small subset of variables in the graph are relevant to the outcomes of interest. Consequently, causal estimation with the full causal graph -- particularly given limited data -- could lead to numerous falsely discovered, spurious variables that exhibit high correlation with, but exert no causal impact on, the target outcome. In this paper, we propose learning a class of necessary and sufficient causal graphs (NSCG) that exclusively comprises causally relevant variables for an outcome of interest, which we term causal features. The key idea is to employ probabilities of causation to systematically evaluate the importance of features in the causal graph, allowing us to identify a subgraph relevant to the outcome of interest. To learn NSCG from data, we develop a necessary and sufficient causal structural learning (NSCSL) algorithm, by establishing theoretical properties and relationships between probabilities of causation and natural causal effects of features. Across empirical studies of simulated and real data, we demonstrate that NSCSL outperforms existing algorithms and can reveal crucial yeast genes for target heritable traits of interest.

We study pointwise estimation and uncertainty quantification for a sparse variational Gaussian process method with eigenvector inducing variables. For a rescaled Brownian motion prior, we derive theoretical guarantees and limitations for the frequentist size and coverage of pointwise credible sets. For sufficiently many inducing variables, we precisely characterize the asymptotic frequentist coverage, deducing when credible sets from this variational method are conservative and when overconfident/misleading. We numerically illustrate the applicability of our results and discuss connections with other common Gaussian process priors.

Discovering causal relationship using multivariate functional data has received a significant amount of attention very recently. In this article, we introduce a functional linear structural equation model for causal structure learning when the underlying graph involving the multivariate functions may have cycles. To enhance interpretability, our model involves a low-dimensional causal embedded space such that all the relevant causal information in the multivariate functional data is preserved in this lower-dimensional subspace. We prove that the proposed model is causally identifiable under standard assumptions that are often made in the causal discovery literature. To carry out inference of our model, we develop a fully Bayesian framework with suitable prior specifications and uncertainty quantification through posterior summaries. We illustrate the superior performance of our method over existing methods in terms of causal graph estimation through extensive simulation studies. We also demonstrate the proposed method using a brain EEG dataset.

The digital twin concept represents an appealing opportunity to advance condition-based and predictive maintenance paradigms for civil engineering systems, thus allowing reduced lifecycle costs, increased system safety, and increased system availability. This work proposes a predictive digital twin approach to the health monitoring, maintenance, and management planning of civil engineering structures. The asset-twin coupled dynamical system is encoded employing a probabilistic graphical model, which allows all relevant sources of uncertainty to be taken into account. In particular, the time-repeating observations-to-decisions flow is modeled using a dynamic Bayesian network. Real-time structural health diagnostics are provided by assimilating sensed data with deep learning models. The digital twin state is continually updated in a sequential Bayesian inference fashion. This is then exploited to inform the optimal planning of maintenance and management actions within a dynamic decision-making framework. A preliminary offline phase involves the population of training datasets through a reduced-order numerical model and the computation of a health-dependent control policy. The strategy is assessed on two synthetic case studies, involving a cantilever beam and a railway bridge, demonstrating the dynamic decision-making capabilities of health-aware digital twins.

Training data attribution (TDA) techniques find influential training data for the model's prediction on the test data of interest. They approximate the impact of down- or up-weighting a particular training sample. While conceptually useful, they are hardly applicable to deep models in practice, particularly because of their sensitivity to different model initialisation. In this paper, we introduce a Bayesian perspective on the TDA task, where the learned model is treated as a Bayesian posterior and the TDA estimates as random variables. From this novel viewpoint, we observe that the influence of an individual training sample is often overshadowed by the noise stemming from model initialisation and SGD batch composition. Based on this observation, we argue that TDA can only be reliably used for explaining deep model predictions that are consistently influenced by certain training data, independent of other noise factors. Our experiments demonstrate the rarity of such noise-independent training-test data pairs but confirm their existence. We recommend that future researchers and practitioners trust TDA estimates only in such cases. Further, we find a disagreement between ground truth and estimated TDA distributions and encourage future work to study this gap. Code is provided at https://github.com/ElisaNguyen/bayesian-tda.

Imitation learning uses data for training policies to solve complex tasks. However, when the training data is collected from human demonstrators, it often leads to multimodal distributions because of the variability in human actions. Most imitation learning methods rely on a maximum likelihood (ML) objective to learn a parameterized policy, but this can result in suboptimal or unsafe behavior due to the mode-averaging property of the ML objective. In this work, we propose Information Maximizing Curriculum, a curriculum-based approach that assigns a weight to each data point and encourages the model to specialize in the data it can represent, effectively mitigating the mode-averaging problem by allowing the model to ignore data from modes it cannot represent. To cover all modes and thus, enable diverse behavior, we extend our approach to a mixture of experts (MoE) policy, where each mixture component selects its own subset of the training data for learning. A novel, maximum entropy-based objective is proposed to achieve full coverage of the dataset, thereby enabling the policy to encompass all modes within the data distribution. We demonstrate the effectiveness of our approach on complex simulated control tasks using diverse human demonstrations, achieving superior performance compared to state-of-the-art methods.

The bulk of the research effort on brain connectivity revolves around statistical associations among brain regions, which do not directly relate to the causal mechanisms governing brain dynamics. Here we propose the multiscale causal backbone (MCB) of brain dynamics shared by a set of individuals across multiple temporal scales, and devise a principled methodology to extract it. Our approach leverages recent advances in multiscale causal structure learning and optimizes the trade-off between the model fitting and its complexity. Empirical assessment on synthetic data shows the superiority of our methodology over a baseline based on canonical functional connectivity networks. When applied to resting-state fMRI data, we find sparse MCBs for both the left and right brain hemispheres. Thanks to its multiscale nature, our approach shows that at low-frequency bands, causal dynamics are driven by brain regions associated with high-level cognitive functions; at higher frequencies instead, nodes related to sensory processing play a crucial role. Finally, our analysis of individual multiscale causal structures confirms the existence of a causal fingerprint of brain connectivity, thus supporting from a causal perspective the existing extensive research in brain connectivity fingerprinting.

Learning curve extrapolation aims to predict model performance in later epochs of training, based on the performance in earlier epochs. In this work, we argue that, while the inherent uncertainty in the extrapolation of learning curves warrants a Bayesian approach, existing methods are (i) overly restrictive, and/or (ii) computationally expensive. We describe the first application of prior-data fitted neural networks (PFNs) in this context. A PFN is a transformer, pre-trained on data generated from a prior, to perform approximate Bayesian inference in a single forward pass. We propose LC-PFN, a PFN trained to extrapolate 10 million artificial right-censored learning curves generated from a parametric prior proposed in prior art using MCMC. We demonstrate that LC-PFN can approximate the posterior predictive distribution more accurately than MCMC, while being over 10 000 times faster. We also show that the same LC-PFN achieves competitive performance extrapolating a total of 20 000 real learning curves from four learning curve benchmarks (LCBench, NAS-Bench-201, Taskset, and PD1) that stem from training a wide range of model architectures (MLPs, CNNs, RNNs, and Transformers) on 53 different datasets with varying input modalities (tabular, image, text, and protein data). Finally, we investigate its potential in the context of model selection and find that a simple LC-PFN based predictive early stopping criterion obtains 2 - 6x speed-ups on 45 of these datasets, at virtually no overhead.